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There's an excellent paper that I've read a few times called "Expanding Confusion" (2004) by Davis and Lineweaver that explains the variety of cosmic horizons quite well. Link to it here.

In section 4.2 of that paper, when they derive a special relativistic and $v=cz$ interpretation for cosmic redshift (and disprove the SR interpretation by 23 sigma), it seems there are potentially some calculation errors: I'm unable to reproduce their results for the apparent magnitude in the B-band $m_B$.

Writing their method out explicitly we have Hubble’s law:

$H = v/D,$

which is added to the longitudinal relativistic Doppler shift in terms of velocity,

$v = c \frac{(1+z)^2 - 1}{(1+z)^2 + 1},$

like so,

$D(z) = \frac{c}{H} \frac{(1+z)^2 - 1}{(1+z)^2 + 1}.$

Then this proper distance is converted to luminosity distance, $D(z)(1+z) = D_L(z)$, whose value we then plug into the distance modulus they used:

$m_B(z)=5log(H_0D_L) +M_B,$

where absolute magnitude $M_B$ = -3.45.

In the v = cz case, they use this for luminosity distance and put it into the same distance modulus above to get their measurements:

$D_L(z)= \frac{cz(1+z)}{H}.$

The errors become clear after a quick calculation: if we input $z = 1$ and $H = 70km/s/Mpc$ for instance, we get $m_B = 24.33$ for the SR interpretation and 25.44 for the $v = cz$ interpretation rather than $m_B = 22.83, 23.94$, respectively, as written in the paper. I've put the corrected magnitude-redshift curves into their original Figure 5.

Did I misunderstand something or was there an oversight?

enter image description here

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    $\begingroup$ This really is a conceptual question (about whether the cooncepts the OP has imbedded in equations are accurate renditions of the concepts in the cited paper). I think it should bd reopened. $\endgroup$ Commented Oct 21, 2024 at 0:34

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In section 4.2, they write

[S]ince SR does not provide a technique for incorporating acceleration into our calculations for the expansion of the universe, the best we can do is assume that the recession velocity, and thus Hubble’s constant, are approximately the same at the time of emission as they are now.

This seems to contradict itself, since a constant recession velocity would be constant $\dot a$, not constant $H=\dot a/a$. If you suppose it's the former that they really meant, then $H = H_0/a = H_0(1{+}z)$. The extra factor of $1{+}z=2$ would shift the curve by $5\log_{10} 2 \approx 1.505$, and your values differ from theirs by $1.50$, so this is probably the culprit.

(Their calculation still doesn't make sense, because cosmological recession speed isn't the same as special-relativistic speed even in a special-relativistic model. They should have used the arctanh of the special-relativistic speed. See this answer.)

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  • $\begingroup$ thanks for the reply, and confirming my suspicion. I've reached out to the authors about it but haven't heard back yet $\endgroup$ Commented Oct 21, 2024 at 21:35

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